integration on inverse trig

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January 31st 2008, 04:21 PM
akhayoon

integration on inverse trig

$\int tan^{-1}u du$

is equal to

$utan^{-1}u-\frac{1}{2} ln(1+u^{2})$

can somebody explain to me how this is true?
January 31st 2008, 04:50 PM
Plato

Quote:

Originally Posted by akhayoon

$\int tan^{-1}u du$ is equal to
$utan^{-1}u-\frac{1}{2} ln(1+u^{2})$ can somebody explain to me how this is true?

Did you differentiate the answer?
Do it!
That shows you how it works.
January 31st 2008, 04:55 PM
akhayoon

well thats the thing I haven't been taught these

it's just given to me as a formula...but I would like to understand how it happened so that way I dont have to memorize it...

thanks
January 31st 2008, 05:02 PM
Plato

Quote:

Originally Posted by akhayoon

well thats the thing I haven't been taught these
it's just given to me as a formula...but I would like to understand how it happened so that way I dont have to memorize it...

Sorry to say, but I find that notion ridiculous.
January 31st 2008, 05:07 PM
akhayoon

well..I really don't care you how you find that notion...all I really want is a hint on how to begin solving that problem really, which is what I always get when I have a question on this forum.
January 31st 2008, 05:46 PM
topsquark

Quote:

Originally Posted by akhayoon

$\int tan^{-1}u du$

is equal to

$utan^{-1}u-\frac{1}{2} ln(1+u^{2})$

can somebody explain to me how this is true?

Quote:

Originally Posted by Plato

Did you differentiate the answer?
Do it!
That shows you how it works.

Here's your hint, as Plato suggested:
$\frac{d}{du} \left ( u~tan^{-1}(u)-\frac{1}{2} ln(1+u^{2}) \right )$

$= tan^{-1}(u) + \frac{u}{u^2 + 1} - \frac{1}{2} \left ( \frac{1}{1 + u^2} \cdot 2u \right )$

What does this tell you?

-Dan
January 31st 2008, 08:20 PM
Jhevon

the responses given seem to me to be geared towards verifying that the claim is true, rather than saying why it's true, or how it works. i believe what the poster was after is actually integrating arctan to get the result shown. in that case, we use integration by parts with $u = \arctan x$ and $dv = 1$ (you do know what i am referring to when i say $u \mbox{ and } dv$ , right akhayoon?)
January 31st 2008, 08:24 PM
Krizalid

A similar integration by parts reasoning is useful here: take the following function $f(u)=u\arctan u.$ Contemplate its derivative and integrate, the rest follows.